NASA has had a series of development projects over the decades.
Most recently the COTS (Commercial Orbital Transportation Service) mainly for developing cargo capacity had a D variant that was never really excersized.
The manned component morphed into CCDev, CCiCAP, and CCtCAP which has been working with SpaceX (Dragon/DragonRider), Sierra Nevada Corp (SNC Dream Chaser) and Boeing (CST-100).
Dragon and CST-100 are capsules, and Dream Chaser is a lifting body designed for landing on a runway. Dragon intends to land under power on land (using the unused launch abort SuperDraco thrusters). CST-100 plans to land on airbags on land.
All three are making progress along their milestones, but all are several years out from regular manned launches. (2020 update: 2020 was the correct year for SpaceX, CST-100 maybe).
In some ways SpaceX may be closest as they have launched and re-entered Dragon at least 5 times unmanned. (2020 update: 20+ times now)
SNC has done an unmanned drop test to practice landing. (Had an issue with one of the landing gear not deploying so it flipped, but met rest of milestones, and the landing gear was not flight equipment on the test).
CST-100 has done a couple of parachute drop tests to test the airbag landing system they intend to use for landing on land.
SNC and Boeing plan to launch on an Atlas-V with unspecified upgrades for man rating. which is interestingly in question now, with the Ukraine/Russia/RD-180 issue.
Dragon of course is flying on a Falcon 9 booster, 4 flights of the F9 1.1 variant at time of this writing, (Update 2020: almost 90 flights) and the base capsule has proven its chops on reentry in 5 missions so far (Update 2020: 20+ missions). They have re-tested their parachute deploy system to validate performance during an abort case (Needs much faster deployment in abort case). They have two launch abort tests due in 2014 (pad abort, and Max-Q abort).
Other than Soyuz, only manned spacecraft in active use would then be Shenzou but that seems even less likely. It is unclear if they have a compatible docking system.
The actual docking system for the proposed new vehicles is due to be launched on a Cargo Dragon in 2015 for installation on the forward facing PMA-2 to convert it to a NDS (NASA Docking System) format. Current cargo craft (JAXA HTV, SpaceX Dragon, and Orbital Cygnus) berth to a CBM, which requires on station intervention to berth and unberth. Docking to a PMA (via NDS/LIDS system) is meant to be possible without people on station.
Since this answer was originally written, many things have changed. Dream Chaser has dropped out of the race.
The Dragon V2/Crew Dragon flew a demo mission unmanned, a full abort, and a 2 person crew to the ISS.
The IDA's were delivered and installed and the PMA modules moved as needed.
NASA has agreed to allow the use of previously flown boosters and capsules on future crew missions.